Classifying Equilibrium Points -- Reference

If this is your first encounter with classifying equilibrium points we recommend that you work through the discovery module before looking at this reference module.

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

This module has two important theorems.

• The Linear Classification Theorem

• The Nonlinear Classification Theorem

If m is not one then the linear dynamical system

p(n + 1) = m p(n) + b

has exactly one equilibrium point: b/(1 - m).

If |m| < 1 then the equilibrium point is attracting and for any initial condition

Proof:

The key idea behind the proof is comparing the distance

to the distance

as follows

This implies that

and

Now, if |m| < 1 then

So,

This completes the proof.

p(n + 1) = m p(n) + b

with different values of m to help see geometrically why this theorem is true.

Suppose that q is an equilibrium point of the dynamical system

p(n + 1) = f(p(n))

Then if |f'(q)| < 1 the equilibrium point q is attracting in the sense that if p(1) is "close to" q then

Proof:

• In the linear case there is exactly one equilibrium point. In the nonlinear case there may be many equilibrium points.

• In the linear case the slope m determines whether the equilibrium point is attracting. In the nonlinear case the derivative at the point q plays the role of the slope.

• The linear theorem is global -- if the equilibrium point is attracting then it "pulls in" the sequence no matter what the initial condition is. The nonlinear classification theorem is local -- if an equilibrium point, q, is attracting then it "pulls in" the sequence if the initial condition is sufficiently close to q. We have seen many examples of this. For example, many pack hunter models have two competing attracting equilibrium points, one of which is zero and another which is nonzero. These two are separated by another repelling equilibrium point which represents a biological threshold. If the initial population is above this threshold then (unless it is very large) it is usually attracted to the nonzero attracting equilibrium but if the initial population is below this threshold then the species dies out -- that is, the sequence approaches the zero equilibrium point.

 
         1.5 p / 500,       if p < 500;
R(p) = {  
         2 - p/1000,        if 500 <= p.

p(n + 1) = R(p(n)) p(n)

Check Your Understanding

1. Consider the logistic model

   p(n + 1) = a [1 - b p(n)] p(n)
   

   Find all its equilibrium points. For which values of a is the zero equilibrium point attracting? For which values of a is the nonzero equilibrium point attracting?

   answer